MATHEMATICA BOHEMICA, Vol. 125, No. 4, pp. 465-479 (2000)

On the equation $\phi (|x^m-y^m|)=2^n$

Florian Luca

Florian Luca, FSP/Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany, e-mail: fluca@Mathematik.Uni-Bielefeld.de

Abstract: In this paper we investigate the solutions of the equation in the title, where $\phi $ is the Euler function. We first show that it suffices to find the solutions of the above equation when $m=4$ and $x$ and $y$ are coprime positive integers. For this last equation, we show that aside from a few small solutions, all the others are in a one-to-one correspondence with the Fermat primes.

Keywords: Euler function, Fermat primes

Classification (MSC2000): 11A25, 11A51, 11A63

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